(4m+5n)2+(5m+4n)2+(4m+5n)(4m-5n)

Jasonbradley

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Jun 16, 2024

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Expanding and Simplifying the Expression (4m+5n)2+(5m+4n)2+(4m+5n)(4m-5n)

This article will guide you through the process of expanding and simplifying the given algebraic expression: (4m+5n)2+(5m+4n)2+(4m+5n)(4m-5n).

Understanding the Expression

The expression involves:

• Squaring binomials: (4m+5n)2 and (5m+4n)2
• Multiplying binomials: (4m+5n)(4m-5n)

Expanding the Expression

1. Expanding the squares:

• (4m+5n)2 = (4m+5n)(4m+5n)

  • Using the FOIL method:
    • First: (4m)(4m) = 16m²
    • Outer: (4m)(5n) = 20mn
    • Inner: (5n)(4m) = 20mn
    • Last: (5n)(5n) = 25n²
    • Combine like terms: 16m² + 40mn + 25n²

• (5m+4n)2 = (5m+4n)(5m+4n)

  • Using the FOIL method:
    • First: (5m)(5m) = 25m²
    • Outer: (5m)(4n) = 20mn
    • Inner: (4n)(5m) = 20mn
    • Last: (4n)(4n) = 16n²
    • Combine like terms: 25m² + 40mn + 16n²

2. Expanding the product:

• (4m+5n)(4m-5n)
  • Using the difference of squares pattern (a+b)(a-b) = a² - b²:
    • (4m)² - (5n)² = 16m² - 25n²

Combining the Expanded Terms

Now, we combine the expanded terms from each part:

(16m² + 40mn + 25n²) + (25m² + 40mn + 16n²) + (16m² - 25n²)

Simplifying the Expression

Finally, we combine like terms to simplify the expression:

16m² + 25m² + 16m² + 40mn + 40mn + 25n² + 16n² - 25n²

= 57m² + 80mn + 16n²

Conclusion

Therefore, the simplified form of the expression (4m+5n)2+(5m+4n)2+(4m+5n)(4m-5n) is 57m² + 80mn + 16n².